Some brat comparisons

Carl wanted these, or at least I think that is what he was asking for.

The meantone fifth with a brat of -1, the Wilson fifth, is the postive
real root of f^4-2f-2=0. Nearby meantones include 50-et, with a brat of
-2.3566, the 81-et, with a brat of -6.0115, and the 88-et with a brat
of -0.6677. Posibly too close to the Wilson is 69-et, with a brat of
-1.0498.

The schismic fifth with a brat of 2 is a root of 5f^9-10f^8+64=0; the
fifth with a brat of 1 is the real root of 5f^9-5f^8-64=0. Nearby
schismic ets are 53, with a brat of 1.4273, 171 with a brat of
0.63723, and 118 with a brat of 4.5724. I wonder if anyone would find a
plot of brat vs fifth in cents for various temperaments interesting?

At 07:05 PM 12/6/2003, you wrote:
>Carl wanted these, or at least I think that is what he was asking for.
>
>The meantone fifth with a brat of -1, the Wilson fifth, is the postive
>real root of f^4-2f-2=0. Nearby meantones include 50-et, with a brat of
>-2.3566, the 81-et, with a brat of -6.0115, and the 88-et with a brat
>of -0.6677.

Great; maybe I'll do an Audio Compositor comparison.

>Posibly too close to the Wilson is 69-et, with a brat of
>-1.0498.

I use 69-tET to approximate metameantone!

>The schismic fifth with a brat of 2 is a root of 5f^9-10f^8+64=0;
>the fifth with a brat of 1 is the real root of 5f^9-5f^8-64=0.

Dear me, I haven't the foggiest how to find the roots of a 9th degree
polynomial.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Dear me, I haven't the foggiest how to find the roots of a 9th
degree
> polynomial.

You own both Maple and Mathematica and can't find the roots of a 9th
degree polynomial? In Maple use the function "fsolve".

>> Dear me, I haven't the foggiest how to find the roots of a 9th
>> degree polynomial.
>
>You own both Maple and Mathematica and can't find the roots of
>a 9th degree polynomial? In Maple use the function "fsolve".

I was thinking they might, but then I seemed to remember something
you wrote about there being no general method for degree > 5, and
the existing methods being spotty.

Ok, so using fsolve I get

-1.189687057, 1.499661576, 1.934825943 for 5f^9-10f^8+64=0

and

1.499846124 for 5f^9-5f^8-64=0.

How do I tune triads with these numbers?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Dear me, I haven't the foggiest how to find the roots of a 9th
> >> degree polynomial.
> >
> >You own both Maple and Mathematica and can't find the roots of
> >a 9th degree polynomial? In Maple use the function "fsolve".
>
> I was thinking they might, but then I seemed to remember something
> you wrote about there being no general method for degree > 5, and
> the existing methods being spotty.

I was talking about exact solutions using radicals--nth roots.

> Ok, so using fsolve I get
>
> -1.189687057, 1.499661576, 1.934825943 for 5f^9-10f^8+64=0
>
> and
>
> 1.499846124 for 5f^9-5f^8-64=0.
>
> How do I tune triads with these numbers?

These are schismic fifths, so the major thirds are t = 32/f^8.
> -Carl

>I was talking about exact solutions using radicals--nth roots.

Can you give an example of such a hard-to-do equation?

>> Ok, so using fsolve I get
>>
>> -1.189687057, 1.499661576, 1.934825943 for 5f^9-10f^8+64=0
>>
>> and
>>
>> 1.499846124 for 5f^9-5f^8-64=0.
>>
>> How do I tune triads with these numbers?
>
>These are schismic fifths, so the major thirds are t = 32/f^8.

Swell. Time to dig out Audio Compositor. Actually, maybe I
should do these in Cool Edit...

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I was talking about exact solutions using radicals--nth roots.
>
> Can you give an example of such a hard-to-do equation?

Any fifth degree polynomial with integer coefficents for which the
Maple function "galois" returns S5 or A5 will do (and that means most
of them.) An example would be x^5-x-1=0, but pick a polynomial at
random, try to factor or use "irreduc" to check if it is irreducible,
and the run it through galois and you're in business.

I'm afraid I don't know a musical application, though the permutation
groups we are talking about on tuning-math are the same kind of
groups as permuations of the roots of an equation.